What about a mismatch in sampling-frequency between Audacity & the sound-card, so it’s unintentionally functioning as a frequency-divider , ( like those bat-detectors which make ultrasonic audible ).
It looks like the “clarinet” artifact frequency is ~1/10th the frequency of the sibilance corresponding with it …
If that was the case, the problem is occurring continuously, but only becomes conspicuous when loud high-frequency sound occurs. ( 1/10th of 200Hz is 20Hz which is inaudible , 1/10th of 7000Hz is 700Hz which is audible ).
If that is the problem the solution would be to make sure the sound-card & Audacity are operating at the same sample-rate.